Join and Meet

Join And Meet

In mathematics, join and meet are dual binary operations on the elements of a partially ordered set. A join on a set is defined as the unique supremum (least upper bound) with respect to a partial order on the set, provided a supremum exists. A meet on a set is defined as the unique infimum (greatest lower bound) with respect to a partial order on the set, provided an infimum exists. If the join of two elements with respect to a given partial order exists then it is always the meet of the two elements in the inverse order, and vice versa.

Usually, the join of two elements x and y is denoted x ∨ y, and the meet of x and y is denoted x ∧ y.

Join and meet can be abstractly defined as commutative and associative binary operations satisfying an idempotency law. The two definitions yield equivalent results, except that in the partial order approach it may be possible directly to define joins and meets of more general sets of elements.

A partially ordered set where the join of any two elements always exists is a join-semilattice. A partially ordered set where the meet of any two elements always exists is a meet-semilattice. A partially ordered set where both the join and the meet of any two elements always exist is a lattice. Lattices provide the most common context in which to find join and meet. In the study of complete lattices, the join and meet operations are extended to return the least upper bound and greatest lower bound of an arbitrary set of elements.

In the following we dispense discussing joins, because they become meet when considering the reverse partial order, thanks to duality.